Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu, a recently developed formal system of proof (presented in Mumma (Synthese 175:255–287, 2010)) within which Euclid’s diagrammatic proofs can be represented.